Abstract

The interior regularity problem for the Leray weak solutions u of the Navier–Stokes equations in a domain Ω ⊂ Rn with n ⩾ 3 is investigated. It is shown that u is regular in a neighbourhood of a point ( x , t ) ∈ Omega; × (0, T ) if there exist constants 0 ⩽ θ < 1 and small ∈ 0 such that with Q 1/ k i ( x , t ) = { x ∈ R n ; | x – x < 1/ k } ( t –1/ k 2, t + 1/ k 2). If ( x , t ) is an irregular point of u , there exists a sequence of non–zero measure sets E ki ⊂ Q 1/ ki ( x , t ) for i = 1,2,..., such that the blow–up rate estimate holds.

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